Image restoration from transformed component data

ABSTRACT

An original image or other data sequence is modified with an internal marker value, and is communicated (stored and/or transmitted) as one component of a transform having multiple complementary components. The image is restored in a closed form, without iteration, and in some cases without any loss of precision from the original image. The original image may have complex element values. Multiple original images may be combined, and separated after restoration. Several types of maps may be embedded in or packaged with the communicated component. Communicated component elements may be stored with variable lengths.

CLAIM OF PRIORITY

[0001] This application claims priority under 35 USC 119(e) from U.S. Provisional Application Ser. No. 60/467,862, filed May 5, 2003, which application is incorporated herein by reference.

TECHNICAL FIELD

[0002] The present invention relates to image data processing, and more particularly concerns restoring an image from a transformed version that includes less than all components of the transformed data.

BACKGROUND

[0003] Digitized images require the storage and communication of large amounts of data. Medical imaging, graphics for games, military reconnaissance, space-based astronomy, and many other applications increasingly strain storage capacity and transmission bandwidth.

[0004] Although image compression techniques exist, some images, for example computerized tomography (CT) medical images require high resolution, high accuracy, and low contrast. Retrieval of these types of images must be lossless, or nearly so. Even less demanding applications are always open to further data compression.

[0005] A number of sources produce images in a transform domain, such as the frequency domain. For example, magnetic resonance imaging (MRI) scans, ultrasound scans, and computed tomography/microscopy devices output magnitude/phase data rather than spatial pixels to represent an image. Transform-domain images produce large amounts of data. For example, a spatial image 1000 pixels square, with 8-bit grayscale resolution requires one megabyte of data. A typical double-precision Fourier transform of that image produces about 16 MB of data: 8 MB for the magnitude component and 8 MB for the phase component. The frequency or magnitude component carries some of the information corresponding to the underlying spatial-domain image, and the complementary phase component carries the rest of the information. The ability to restore or reconstruct a spatial-domain image from only one of these transform-domain components could halve the bandwidth and/or time required to transmit the full image in either the spatial or transform domain. Storing only one component could reduce by 50% the space requirements on a disk or other medium. The U.S. Food and Drug Agency (FDA) requires that all medical source data has to be stored. Even when an image is generated initially in the spatial domain, it may be desirable to store or transmit the image in a transform-domain form.

[0006] Copending commonly assigned U.S. patent application Ser. No. 10/124,547, filed Apr. 17, 2002, demonstrates an iterative technique for restoring a spatial-domain image transmitted or stored as a single component of a transform-domain representation. Unlike previous such techniques, it requires no special conditions in the original image. However, being iterative, it requires processing time and capacity to pursue a number of iterations, making it hundreds or even thousands of times more computation intensive than a non-iterative or closed-form technique. Also, restoration is not lossless; the solution can only approach the pixel values of the original image more or less closely.

SUMMARY

[0007] The present invention offers methods for modifying an original image, communicates a single component of the image in a transform domain, and restores the original image in closed-form, non-iterative processes. The invention encompasses methods, systems, and media.

DRAWING

[0008]FIG. 1 is a block diagram of a system including different aspects of the invention, and includes diagrams of image data processed by the system.

[0009]FIG. 2 is a flowchart showing an example of a method for modifying an original image for restoration.

[0010]FIG. 3 shows an example computer program for modifying an original image for restoration.

[0011]FIG. 4 is a flowchart showing an example of a method for restoring a modified image from a single transform-domain component.

[0012]FIG. 5 shows a computer program for restoring a modified image from a single transform-domain component.

[0013]FIGS. 6A-6C present a demonstration that image restoration is lossless.

[0014]FIG. 7 is a flowchart of another example method for modifying an original image.

[0015]FIG. 8 shows another example computer program for modifying an original image.

[0016]FIG. 9 is a flowchart of another example method for restoring a modified image.

[0017]FIG. 10 shows another example program for restoring a modified image from a single transform-domain component.

[0018]FIG. 11 is a flowchart of a further example method for modifying an original image for restoration.

[0019]FIG. 12 shows a further example program for modifying an original image.

[0020]FIG. 13 is a flowchart of a further example method for restoring a modified image.

[0021]FIG. 14 shows a further example program for restoring a modified image.

[0022]FIG. 15 is a flowchart of an additional example method for modifying one or more original images.

[0023]FIG. 16 is a flowchart of an additional example method for restoring one or more modified images.

[0024]FIG. 17 shows an application of the methods of FIGS. 15 and 16.

DETAILED DESCRIPTION

[0025]FIG. 1 is a high-level schematic of an embodiment of a system 100 capable of hosting the invention. System 100 may be implemented in hardware, software, or a combination of both. Any of a number of source devices 110, such as magnetic-resonance imaging (MRI) scanner 111, ultrasound scanner 112, or satellite scanner 113 may produce the original image. Satellite water-vapor scanners, for example, produce frequency-domain images. Additionally, the original image may be retrieved from a storage device 114. A frequency-transform domain original image from one of the input devices 110 is represented as X(1 . . . K₁,1 . . . K₂), having both magnitude and phase values 1 through K₁ in a first dimension and 1 through K₂ in a second dimension. Transform image X corresponds to an underlying original image in a spatial or other base domain, which is denoted as X(1 . . . N₁, 1 . . . N₂) for an N₁×N₂ pixel image. In this example, the base-domain data points must have real values. However, complex-valued data can be treated as two sequences each having real values.

[0026] Although two-dimensional images are common, system 100 can process images in any number of dimensions. In particular, a two-dimensional image is usually treated as N (N=N₁ or N₂) separate one-dimensional pixel sequences X ₁(1 . . . K₁) . . . XN(1 . . . K₂). Because the Fourier transform is separable, a two-dimensional image can be decomposed by rows or by columns, each row or column processed with a one-dimensional transform, and the N results stored as a two-dimensional transform X(1 . . . K₁,1 . . . K₂). This method offers considerable computational savings over direct two-dimensional processing.

[0027] The K₁×K₂ values of the transform-domain image components X are rational values to produce a lossless restoration. In a digital system, all representable values have a finite precision, and thus are effectively rational. Since all rational values can be considered as integers simply by changing representation scale, the terms “rational” and “integer” are treated as equivalent herein.

[0028] Processor 120 receives input image X over bus 121, and stores it in a storage facility 122, which may include internal storage and/or a removable medium for holding image X and program instructions for transforming and manipulating the image. Storage 122 also holds a marker quantity C, which may comprise a single complex value, as explained below. Storage 122 further holds a modified transformed form XC of image X. If the transform is a Fourier transform, XC has two complementary components, a magnitude MXC(1 . . . K₁,1 . . . K₂) matrix and a phase PXC(1 . . . K₁,1 . . . K₂) matrix. Only one of these components, the magnitude MXC, is output to a communication medium 130. The complementary phase component PXC may be discarded. The term “communication” has a broad meaning herein; device 130 may include a medium for transmitting the image component, and/or for storing it for later restoration. In this embodiment, MXC is treated as the modified magnitude component of a discrete Fourier transform (DFT) of a two-dimensional spatial image X(1 . . . N₁,1 . . . N₂). Each pixel of the spatial image may represent a gray-scale amplitude, one or more color intensities, or any other characteristic, in any desired format.

[0029] Processor 120 sends the image magnitude component MXC to medium 130 for storage and/or transmission to processor 140. The magnitude component may be further encoded for storage or transmission if desired. An advantage of the present technique is its transparency to any form of lossless compression or encryption. For example, a lossless Lempel-Ziv-Welch (LZW) compression algorithm can be applied to MXC, so as to achieve an additional 50% reduction of data volume.

[0030] Processor 140 receives transformed image component after transmission by and/or storage in medium 130. Processor 140 inverse transforms the communicated component, and removes a marker C in the base domain. marker C is equivalent to marker C in the transform domain, as explained below. Marker value(s) may be transmitted between the processors along with component MXC, or may be made known to both processors in other ways. One possible way to communicate the marker value is to embed it within MXC, in one or more locations where the data points are known to have very small values, such as the high-frequency DFT coefficients. For security, the marker location may be hidden, or it may be scrambled over several locations, making reconstruction difficult without a key. Difficulty of detection is enhanced if the real/imaginary or magnitude/phase values of the marker are embedded so as to maintain symmetry properties of the DFT.

[0031] Memory 142 stores a base-domain form C of embedded marker C along with the final restored base-domain image QF(1 . . . N₁,1 . . . N₂), as well as program instructions for restoring the image, as in any conventional processor. Processor 140 outputs final image Q on bus 141 to output devices 150. Examples of output devices include a display 151, a printer 152, and a data analyzer for further processing of the restored image.

[0032] Some applications may desire a transform-domain form of the restored image. Storage 142 includes this form as two arrays representing the magnitude MQ(1 . . . K₁,1 . . . K₂) and phase PQ(1 . . . K₁,1 . . . K₂) of the DFT-domain restored image. Because the phase component of the original transformed image was discarded before transmission or storage of the (modified) magnitude component, this restoration is known as phase retrieval.

[0033]FIG. 1 shows elements 110-122 as separate from elements 140-153. However, some applications may modify and store an image for restoration on the same computer at a later time. In such cases, only a single processor, storage, and/or bus are employed. Although the marker is normally embedded before communication, it could be embedded after the component is received or retrieved. System 100 may be realized as one or more general-purpose computers, as specialized systems, as elements embedded within other equipment, or in any other physical form appropriate for a particular application.

[0034]FIG. 2 is a flowchart of an example method 200 for modifying a transform-domain image, representing an image in a spatial or other base domain, for communication as single component in the transform domain. For continuity of exposition, this example assumes a two-dimensional spatial image X(1 . . . N₁,1 . . . N₂) having pixels expressed as numbers that are rational in terms of a preselected precision. The precision may differ for different embodiments, or for different images in the same embodiment. Standard double-precision floating-point pixel values are assumed for this example, although integer and other floating-point representations are possible.

[0035] Block 210 receives an image from an image-generating device, a storage, or other input. Normally, this is already expressed in the transform domain, either because it is generated in that form or because it has been converted externally to that form. However, if an application prefers to convert it internally from a base-domain form X(1 . . . N₁,1 . . . N₂) to transform-domain X(1 . . . K₁, 1 . . . K₂), block 210 may also perform a conversion by way of a discrete Fourier transform (DFT) or other technique. The number of frequencies K₁ and K₂ in each dimension are selected according to known principles for an appropriate resolution in the restored image. Mathematically, X(1 . . . K₁,1 . . . K₂)=Re[X(1 . . . K₁,1 . . . K₂)]+jIm[X(1 . . . K₁,1 . . . K₂)], the sum of real and imaginary components. These components are also called the Cartesian components of X. In polar coordinates, the components are called the magnitude MX(1 . . . K₁,1 . . . K₂) and the phase PX(1 . . . K₁,1 . . . K₂). The real component is complementary to the imaginary component in that together they represent all of the information in the original base-domain image X; the magnitude component complements the phase for the same reason.

[0036] Block 220 generates a known marker value C. For most applications, the marker is a single complex number, expressed in polar coordinates as a magnitude (or radius) MC and a phase PC such that C=MCexp(jPC), or equivalently in Cartesian form as the sum of real and imaginary parts, C=RC+jIC and is added to every element of X. That is, the modified transform image XC(1 . . . K₁,1 . . . K₂)={Re[X(1 . . . K₁,1 . . . K₂)]+RC}+j{Im[X(1 . . . K₁,1 . . . K₂)+IC]}, in Cartesian form. It is usually more convenient to set the marker radius as a function of the pixel values in the base-domain image X(1 . . . N₁,1 . . . N₂).

[0037] A convenient function for setting the marker radius is MC=A×N₁×N₂, an amplification factor A times the number of pixels in the underlying base-domain image. The pixel number may represent the size of the underlying base-domain image X, or the size or a largest image to be sent by a system, or in some other manner; these numbers can be approximate or estimated. The amplification factor A may be selected primarily from the maximum pixel magnitude 2{circumflex over ( )}q of the underlying image X. Other possible factors include the precision of the numerical representations for the images, and rounding performed later in the process; the size of A may also be approximate or estimated. A safe lower bound lies in the neighborhood of A=2{circumflex over ( )}q×10{circumflex over ( )}2. Another approximate empirical formula is A=10{circumflex over ( )}5/2{circumflex over ( )}q. Other values are also feasible, up to several orders of magnitude higher, and lower to some extent. The amplification factor may be selected individually for each image, or may be fixed at one value for all images in a set or in a system. Maximum radius is not critical; it need only be low enough so as not to cause ambiguity in the pixel values of the restored image. The marker must be at least large enough to guarantee that the modified image is minimum-phase in the transform domain. Maximum value usually depends upon the available software or hardware floating-point precision.

[0038] The marker phase PC can assume any value except those near 0°, 90°, 180°, or 270°. As a practical matter, using standard floating-point processing, the phase should not lie within the range of ±1° from the four forbidden values. This range can be decreased if desired, by increasing the value of the marker radius MC. PC=45° is a convenient approximate value, and may simplify calculation in some instances.

[0039] For some ordinary applications, the marker C remains constant, so that block 220 merely retrieves it from storage for multiple images. Other environments may request block 220 to calculate a separate marker magnitude and/or phase for some or all images.

[0040] Block 230 embeds marker C within image X to produce a modified original image XC. Where C has a single complex value, that value is added to every element of the transform image; in the Cartesian formulation, XC(1 . . . K₁,1 . . . K₂)={Re [X(1 . . . K₁,1 . . . K₂)]+RC}+j{Im[X(1 . . . K₁,1 . . . K₂)]+IC]}, where C is the sum of its real and imaginary components, C=RC+jIC.

[0041] The transform-domain marker in this example represents the transform-domain equivalent of a delta functional in the base domain. Therefore, an base-domain marker C=cδ(n₁,n₂) could alternatively be applied to a pixel at location X(n₁,n₂) of the base-domain image, if that image is available at the source. For an image formed as a sequence of discrete pixels, applying a delta functional amounts to adding a predefined constant value to only one of the existing pixels, conveniently to the first element, such as X(1,1) in a two-dimensional image. More generally, it is possible to define markers as functions other than transform-domain constants or base-domain delta functionals.

[0042] Block 240 calculates the magnitude component of the modified transform-domain image from its real and imaginary parts as MXC(1 . . . K₁,1 . . . K₂)={Σ[ReXC(1 . . . K₁,1 . . . K₂){circumflex over ( )}2+ImXC(1 . . . K₁,1 . . . K₂){circumflex over ( )}2]}{circumflex over ( )}0.5, the square root of sum of the squares. Block 250 sends only the magnitude component MXC to medium 130, FIG. 1, for transmission and/or storage. If the marker C is redetermined for each image, its values may be sent along with the magnitude as, e.g., part of a file descriptor or header, or within the image data itself, as noted above.

[0043]FIG. 3 shows a computer program for carrying out a method 300 for modifying a base-domain image for communication as single component in a transform domain. The program is written in the Matlab® language, publicly available from The Math Works, Inc. The original input image may have any number of dimensions. The variable names are explained in the figure.

[0044]FIG. 4 is a flowchart showing an example of a method 400 for restoring a modified image from a single transform-domain component. Block 410 receives the magnitude component MXC from medium 130, FIG. 1. Block 420 performs an inverse DFT or other transform upon MXC to produce an approximate restored image QA(1 . . . N₁,1 . . . N₂) in the base or spatial domain.

[0045] Blocks 430 correct for the marker added in methods 200 and 300 above. Block 431 separates QA into an even part EQA and an odd part OQA. Mathematically, EQA(1 . . . N₁,1 . . . N₂)=[EQ(1 . . . N₁,1 . . . N₂)+EQ(N₁ . . . 1,N₂ . . . 1)]/2. EQ(N₁ . . . 1,N₂ . . . 1) is generated by reversing EQ(1 . . . N₁,1 . . . N₂) and shifting it circularly one place to the right. The following section gives an example of this procedure; it is a conventional block in most digital signal processor (DSP) integrated circuits. The corresponding odd sequence is OQA(1 . . . N₁,1 . . . N₂)=[OQA(1 . . . N₁,1 . . . N₂)−OQA(N₁ . . . 1,N₂ . . . 1)]/2.

[0046] Block 432 subtracts the base-domain equivalent form C of the marker from the even part. In this example, C represents a delta function to be subtracted from the first pixel location {1,1}, so block 432 performs EQA(1,1)=EQA(1,1)−MC. The even part is then scaled to remove the marker phase, EQA(1 . . . N₁,1 . . . N₂)=EQA(1 . . . N₁,1 . . . N₂)/cosPC. Because the first element of the odd part is zero when pixels have real values, no subtraction need be undertaken, and the odd part must only be scaled for the marker phase PC, by OQA(1 . . . N₁,1 . . . N₂)/sinPC.

[0047] Block 440 resolves the approximate parts to produce exact quantities for restoring the original pixel values from their property of having rational values. The even part is rounded to create an exact or final even part EQ(1 . . . N₁,1 . . . N₂)=Rational[EQA(1 . . . N₁,1 . . . N₂)]. In this example, the function Rational(V)=[Integer(2×10{circumflex over ( )}d×V)]/(2×10{circumflex over ( )}d), where d is the number of decimal places in the representation of its argument V; equivalent functions for binary or other number representations are easily defined. The purpose of this function is to convert a value that lies halfway between two possible values of a pixel in the original image into the correct one of those values; the following section explains why this operation produces exact pixel values. Other functions that achieve this result are also acceptable. Block 440 operates similarly upon the odd part to produce an exact odd part OQ(1 . . . .N₁,1 . . . N₂)=Rational[OQA(1 . . . N₁,1 . . . N₂)].

[0048] Block 450 combines the exact even and odd parts to produce the final restored image

Q(1 . . . N ₁,1 . . . N ₂)=EQ(1 . . . N ₁,1 . . . N ₂)+OQ(1 . . . N ₁,1 . . . N ₂).

[0049] Block 460 then outputs Q as the final restored image having pixel values that are the same as those in the original image X(1 . . . N₁,1 . . . N₂) for a lossless reconstruction.

[0050] Some applications may wish to reconstruct the full transform-domain image in addition to—or even instead of—the base-domain image Q. Blocks 470 carry out this task. Block 471 calculates the real part of the restored transform-domain image

RQ (1 . . . K ₁,1 . . . K ₂)=DFT[EQ(1 . . . N ₁,1 . . . N ₂)

[0051] and the imaginary part

IQ (1 . . . K ₁,1 . . . K ₂)=DFT[OQ(1 . . . N ₁,1 . . . N ₂).

[0052] Block 472 computes the retrieved phase component as

PQ =arctan[ IQ (1 . . . K ₁,1 . . . K ₂)/ RQ (1 . . . K ₁,1 . . . K ₂).

[0053] Because the received magnitude MXC contains the embedded marker value, block 472 computes the true magnitude component

RQ ={[RQ(1 . . . K ₁,1 . . . K ₂)]{circumflex over ( )}2+[ IQ (1 . . . K ₁,1 . . . K ₂)]{circumflex over ( )}2}.

[0054] These calculations could produce polar-coordinate forms instead of the Cartesian forms described. Block 480 outputs the two arrays RQ and IQ or polar-coordinate magnitude and phase, if desired.

[0055]FIG. 5 shows a computer program for carrying out a method 500 for restoring an image communicated as a single component in a transform domain. The program is written in the Matlab® language. The transform image component may have any number of dimensions. The variable names are explained in the figure. The quantity eps in the code is the smallest number representable by the system, to forestall computational artifacts; other implementations may not require it.

Explanation of Operation

[0056] The foregoing describes a representative form of communicating only one component of a transform-domain image, and then restoring the entire image in a closed form—i.e., without iteration. This section demonstrates why the restoration can be lossless. The text refers to numbered equations in FIG. 6. Without loss of generality, this section assumes restoration from a magnitude component, one-dimensional sequences, zero-origin indices, and integer data-sequence values, for clarity of exposition. Complex numbers Z are represented interchangeably in the equivalent Cartesian form Z=ReZ+jImZ and polar or magnitude/phase form Z=(MagZ)×exp(−jPhZ), where j={square root}−1 and “exp” denotes the exponential function.

[0057] Eqn. 601 defines each term X(k) of the forward or analysis discrete Fourier transform (DFT) of an N-point sequence having data values X(n), 0≦n≦N−1. Eqn. 602 shows the inverse or synthesis form of the DFT. Each term X(k) is a complex quantity in the frequency or transform domain, and can be represented in either form of Eqn. 603. R and I prefixes denote complementary real and imaginary components; M and P denote complementary magnitude and phase components.

[0058] A number of transforms, including the DFT, have the linearity property of Eqn. 604, where X₁, X₂ are any two N-point sequences in a base domain, and a, b are simple scalar constants.

[0059] A property of any sequence X is that it can be represented as the sum of an even sequence EX and an odd sequence OX, both having N points. Eqn. 605 defines the terms EX(n) of the even sequence. The notation X(−n) signifies the nth term of a sequence that is reversed and circularly shifted one place to the right from the original sequence X. For example, an original sequence {1, 3, 5, 2, 4, 6} is reversed and shifted to {1, 6, 4, 2, 5, 3}, and its even sequence becomes {1, 4.5, 4.5, 2, 4.5, 4.5} according to Eqn. 605. The DFT of any even sequence is entirely real (i.e., not complex-valued), and equals the real part of the full original sequence, as shown in Eqn. 606. Eqn. 607 defines the odd sequence terms OX(n). The odd sequence in the above example is {0, −1.5, 0.5, 0, −0.5, 1.5}. The DFT of any odd sequence is entirely imaginary, and equals the imaginary part of the original sequence, Eqn. 608.

[0060] Eqns. 605 and 607 demonstrate that, if a sequence has only integer values, then the even and odd sequences consist entirely of integer or half-integer values. Extending this property from integer to rational values reveals that, to any selected scale of a digital number representation, the even and odd sequence values are either exact at that scale or halfway between two exact units.

[0061] The form of a delta functional or impulse for discrete sequences has an amplitude C for only one of the sequence elements—usually taken to be the first element—and is zero for all other elements. The DFT of a delta in the base domain is a complex constant C that is the same for all frequencies k, and has a value given by Eqn. 609. Eqn. 610 shows the polar and Catesian forms of C. MC is the marker magnitude or radius; PC is the marker phase. Because of the linearity property of the DFT, Eqn. 604, adding a delta distribution to a base-domain sequence X, Eqn. 611, is is equivalent to adding the constant C to every term of the sequence's DFT X, Eqn. 612.

[0062] Eqn. 613 calculates the magnitude component MXC for each term of the modified DFT image XC. This magnitude may be stored or transmitted by itself for lossless restoration of the original base-domain image X, or of the entire transform-domain image X.

[0063] Restoration of the original image X from only the component MXC employs the DFT properties of evan and odd sequences shown in Eqns. 606, 608, and the previously noted fact that a function is the sum of its even and odd parts. Eqn. 614 applies an inverse DFT which, by Eqn. 606, produces a sequence of base-domain terms EQAC(n) that represent an even sequence for restoring the original image. Eqn. 615 produces a sequence of terms OQAC(n) representing a corresponding odd sequence in the base or spatial domain.

[0064] Sequences EQAC and OQAC still contain a base-domain form C of the transform-domain marker C added in Eqn. 612. In this example, the transform-domain marker C was applied to correspond to a base-domain delta functional of amplitude C at the first image location; that is, XC(0)=X(0)+C. The marker magnitude only affects the even sequence, and thus need be subtracted only from that part in Eqn. 616. The divisor, cos(PC), removes the known phase PC of the marker. Eqn. 617 removes the marker phase from the odd sequence with divisor sin(PC). Restricting the marker phase away from the vicinity of the zeros of these two divisors prevents them from introducing overrun or divide-by-zero faults in a digital processor.

[0065] For sufficiently large marker values, Eqns. 616 and 617 produce even and odd sequences EQA and OQA that are very close to—but not exactly—the even and odd parts of the restored image. The difference from the true sequence-element values arises from the possible introduction of half-integer values into the even and odd sequences by Eqns. 605 and 607. However, the restriction of the original sequence elements to integer values allows the half-integer items to be found and corrected. Eqns. 618 and 619 produce exact, integer even and odd sequences EQ and OQ from the approximate sequences EQA and OQA respectively. The second form of Eqns. 618 and 619 may be used for pixel values having any finite number d of decimal points.

[0066] Finally, because any function is the sum of its even and odd parts, Eqn. 620 yields the pixel terms of the lossless restored image, Q=X.

[0067] Restoring the magnitude and phase components then follows from the definitions of those components in Eqn. 603. After transforming the restored base-domain image DFT(Q)=RQ+jIQ, Eqn. 621 calculates terms of the restored magnitude component MQ=MX, and Eqn. 622 calculates terms of the phase PQ=PX.

Restoration from Phase Component Only

[0068]FIG. 7 is a flowchart of an example method 700 for modifying an image in a spatial or other base domain for communication as a single phase component in the transform domain, instead of using the magnitude component as in method 200. Block 711 transforms the image to the transform domain, here again by a discrete Fourier transform. Although the marker could have been added in the base domain, block 720 generates or selects a transform-domain marker of value C having a magnitude R, determined in substantially the same way as previously described in connection with marker MC in method 200, FIG. 2, above. The marker is complex, having a phase arbitrarily set to 45°. Block 730 embeds the marker by adding it to every element of the transformed original image. Block 740 calculates the phase component, and block 750 communicates (transmits or stores) only the phase component as a representation of the entire modified image.

[0069]FIG. 8 is a Matlab® program 800 for modifying an image for communication as only a phase component in the transform domain. The code sets a marker value C having a magnitude R and a complex phase phi=45°. An array PXIC(1 . . . K₁,1 . . . K₂), representing the phase elements of the modified image XIC(1 . . . K₁,1 . . . K₂) may then be communicated—i.e., transmitted and/or stored—on a medium such as 130, FIG. 1.

[0070]FIG. 9 is a flowchart of an example method 800 for recovering or restoring the modified image from phase component 750, FIG. 7, shown in FIG. 9 as block 910. The marker magnitude and phase may also be acquired here if desired. Block 920 forms a complex discrete Fourier transform from the phase component, and block 930 performs an inverse transform, producing a base-domain image. Block 940 removes the marker from its known pixel location(s) in the image. Alternatively, the transform-domain version of the known marker could be removed before the inverse transform of block 930. Block 950 calculates the real part of the approximate restored image. Block 960 calculates the exact form of the restored image similarly to block 440, FIG. 4. Here again, the entire transformed version, including the magnitude component, of the restored image could be calculated if desired.

[0071]FIG. 10 shows a Matlab® program 1000 for restoring an image from a phase component, according to method 900. After inverse-transforming this component, the program removes the marker phase phi and magnitude (radius) R from the base-domain form JC. After calculating the real part of the image J(1 . . . N₁, 1 . . . N₂) from which the marker has been removed, the program applies the procedure described in connection with method 200 for producing the exact pixel values of the restored image in the base domain.

Restoration from Sign-Magnitude Component

[0072]FIG. 11 is a flow chart of a method 1100 for modifying a transform-domain image for communication as a single component in the transform domain. This example describes a one-dimensional row or column of an image X(1 . . . N)=ReX(1 . . . N)+jImX(1 . . . N), which may have multiple dimensions. The base-domain pixels are numbers that are rational in terms of a preselected precision, which may differ for different embodiments.

[0073] Block 1110 receives the image from an image-generating device, a storage, or other input. Normally, this is already expressed in the transform domain, as

X (1 . . . K)=ReX(1 . . . K)+jIm X(1 . . . K)= MX (1 . . . K)exp(jPX(1 . . . K)),

[0074] in Cartesian or polar coordinates. However, it could be received in the base domain and transformed in block 1110; the remainder of method 110, and all of method 1300 below, would remain unaffected.

[0075] Block 1120 generates a known marker having a strictly real value of magnitude C. Appropriate values of C may be selected as described in connection with method 200, FIG. 2.

[0076] Block 1130 embeds marker C within image X to produce a modified original image

XC (1 . . . K ₁)={Re [X(1 . . . K ₁)]+ C }+j{Im[X(1 . . . K ₁]}.

[0077] Because C is real, the imaginary part of X remains unchanged. The marker in this example represents the transform-domain equivalent of a delta functional in the base domain. As in method 200, a base-domain marker C=cδ(n₁,n₂) could alternatively be applied to a pixel at location X(n₁,n₂) of the base-domain image, if that image is available at the source. For an image formed as a sequence of discrete pixels, applying a delta functional amounts to adding a predefined constant value to only one of the existing pixels, conveniently to the first element, such as X(1) in a one-dimensional image, or a row or column of a two-dimensional image. Again, markers may be functions other than transform-domain constants or base-domain delta functionals.

[0078] Block 1140 calculates both the magnitude and phase of the modified transform-domain image from its real and imaginary parts. The magnitude component is

MXC (1 . . . K ₁)=[ReXC(1 . . . K){circumflex over ( )}2+ImXC(1 . . . K){circumflex over ( )}2]{circumflex over ( )}0.5.

[0079] The phase component is

PXC =arctan {Im[XC(1 . . . K)])/Re[XC(1 . . . K)]}.

[0080] Block 1150 generates a sign map of XC. This quantity is defined as map(1 . . . K)=1−2×[PXC(1 . . . K)<0]. That is, each element of map has a value of +1 or −1 depending upon whether the corresponding element PXC has a phase that is negative or non-negative. Block 1160 embeds elements of the sign map into corresponding elements of the magnitude component, SMXC(1 . . . K)=map*MXC, where the asterisk denotes term-by-term multiplication.

[0081] Block 1170 then sends only the magnitude component SMXC to medium 130, FIG. 1, for transmission and/or storage, and sends with it the sign map map. Again, if the marker C is redetermined for each image, its value may be sent along with the signed-magnitude component.

[0082]FIG. 12 shows a Matlab® program for an example of a program 1200 for modifying an original image for communication as a single signed-magnitude component in a transform domain. The variable names are noted in the figure.

[0083]FIG. 13 illustrates a method 1300 for restoring an original base-domain image from only a signed-magnitude component. Block 1310 receives the signed magnitude component SMXC(1 . . . K) from medium 130, FIG. 1. Block 1320 extracts the sign map from the sign of the component, map(1 . . . .K)=1−2×[SMXC(1 . . . K)<0]. Block 1330 calculates the component's magnitude

MXC =|SMXC|=[ReSMXC(1 . . . K){circumflex over ( )}2+ImSMXC(1 . . . K){circumflex over ( )}2]{circumflex over ( )}0.5.

[0084] Block 1340 expands the magnitude by one place of accuracy, in this example one decimal place, in the base domain of the image. Inverse transforming, EQA=ifft(MXC). Then the base-domain part acquires an extra place, Integer(2×EQA)/2=EQA. This quantity is then converted back to the transform domain, Re(XC)=fft(EQA).

[0085] Block 1350 restores the phase of the image. The raw phase is calculated as

PXC =arccos[Re( XC )/ MXC ].

[0086] Then the map supplies the actual phase,

PSXC (1 . . . K)= map (1 . . . K)* PXC (1 . . . K).

[0087] Again, the asterisk denotes term-by-term multiplication.

[0088] Block 1360 produces a base-domain image from the entire transform form, QCA=ifft[MXCexp(jPXC)].

[0089] Block 1370 removes the marker as in method 200, QA(1)=QCA−C. When the marker is a delta functional, its value is subtracted only from the base-domain element where it was applied—the first element, in this example.

[0090] Block 1380 produces the exact base-domain image by rounding the approximate form, Q(1 . . . K)=Integer[QA(1 . . . K)]. This image has the exact element values of the original image in its base-domain form. For example, if the transform image received in block 1110 represented a 256-level gray-scale image in the base domain, then restored image Q has exactly the same accuracy. Block 1390 stores or transmits this restored image Q for output or further processing.

[0091]FIG. 14 shows a program 1400 for restoring the original image from a signed magnitude component. In this example, the original input image has two dimensions, and is received in the base domain, so that the original and restored images may be compared.

Complex Image Pixels

[0092] Previously described methods assumed that the pixels in the original image have only real values. Some applications, however, produce pixels that have complex values. Synthetic-aperture radars, seismic analysis, and some medical imaging, for example, produce transform-domain images that translate back to a spatial or other domain as complex-valued pixels. In addition, there are applications in which the ability to process complex pixel values is advantageous even when the actual pixel values are strictly real.

[0093]FIG. 15 is a flow chart of an example method 1500 for modifying such an image for communication as a single component in the transform domain, where the original image-pixel values may assume complex values. Block 1510 represents an original input image having complex pixel values in a base domain. Although the image may have any number of dimensions, this example focuses upon a two-dimensional matrix of pixels I(n₁,n₂)=I₁(n₁,n₂)+j I₂(n₁,n₂), representing the real and imaginary parts. Block 1511 transforms the base-domain image to a transform-domain image X(k₁,k₂)=fft₂(I(n₁,n₂). If the original image arrives in transform-domain form, block 1511 is not needed. Unless otherwise from the context, equations are expressed in the syntax of the aforementioned Matlab® language.

[0094] Block 1520 generates a sign map similarly to block 720, FIG. 7; because this example method employs other maps as well, this map is called a phase map. Block 1520 calculates the phase component PX(k₁,k₂) of the transformed image and constructs a binary-valued map SGN₁(k₁,k₂) having one value (e.g., “1”) where the phase has a negative value, and another value (e.g., “0”) where the sign of the phase is positive.

[0095] Block 1530 generates a real-valued marker, similarly to block 730. An amplification factor may be employed, as mentioned previously, to ensure minimum phase of the modified image in the transform domain. This factor may be based upon the image pixel values, or may be a constant. Typical markers may be C=α N₁N₂, where α=10{circumflex over ( )}4 to 10{circumflex over ( )}8. Block 1540 embeds the marker as described previously, and block 1550 calculates the magnitude component of the thus modified image, MXC(k₁,k₂)=|X(k₁,k₂)+C)|.

[0096] Blocks 1560 may be added to reduce image storage requirements in another way, by reducing the storage required for individual matrix values in the magnitude component. Block 1561 reduces the highest values of the matrix elements by subtracting the large marker value from every element, MXC1(k₁,k₂)=MXC(k₁,k₂)−C. Because the resulting element values may be negative as well as positive, block 1561 then generates a sign map SGN₂(k₁,k₂) having one value (e.g., “1”) for elements having a negative value—i.e., below a threshold value of 0—and another value (e.g., “0”) for positive elements, above 0 or some other threshold. Block 1562 reduces storage by allowing each matrix element of the magnitude component to occupy a variable number of bytes of storage. Block 1562 first computes the absolute values of the individual elements, MXC1(k₁,k₂)=|MXC1(k₁,k₂) |, and rounds them to the nearest integer. Block 1562 then generates a location map or byte-length map

BYTE(k ₁ ,k ₂)=ceiling (|log2( MXC 1(k ₁,k ₂))/8|).

[0097] Each matrix element thus indicates how much storage is required for the corresponding element of MXC1. In this example, each element of BYTE has three bits, indicating that one through eight bytes may be used for each element of MXC1. This allows each of the elements of the magnitude matrix MXC1(k₁,k₂) to be stored with variable byte lengths, rather than requiring the same storage for each element, regardless of its actual magnitude. Storage units other than bytes may be employed here if desired. Although blocks 1560 add overhead, the storage/bandwidth savings may outweigh it in many cases.

[0098] Block 1570 packages the three maps SGN₁(k₁,k₂), SGN₂(k₁,k₂), BYTE(k₁,k₂) along with the magnitude component MXC1 (k₁,k₂). The maps may be embedded in the image itself, sent as separate files, or packaged in any other convenient manner. As before, the value of the marker C (or α. N₁, N₂), as well as other information such as an image name, may be included if desired. Block 1580 represents the total package that is communicated—i.e., transmitted or stored for future restoration.

[0099]FIG. 16 depicts a method 1600 for restoring an original complex-valued base-domain image from a signed-magnitude transform-domain component 1610 such as that communicated in a medium 130, FIG. 1, from block 1580, FIG. 15. Block 1620 extracts the maps SGN₁(k₁,k₂), SGN₂(k₁,k₂), BYTE(k₁,k₂), and possibly other data, packaged along with th magnitude component MXC1. The phase (sign) and magnitude maps may be converted to a form that is more useful in this particular example, MAP₁(k₁,k₂)=1−2*SGN₁(k₁,k₂) and MAP₂(k₁,k₂)=1−2*SGN₂(k₁,k₂). Block 1630 reconstitutes the full magnitude component from the compressed form in which it was communicated to block 1610. Location map BYTE(k₁,k₂) may optionally expand the variable-byte form by known methods to a form having the same number of bytes per element, for easier computation in later stages. The marker value subtracted in block 1561 is restored as MXC(k₁,k₂)=MXC1(k₁,k₂)+C.

[0100] Block 1640 calculates the real part of the restored transform-domain image. The marker is partially removed in the base domain during this process, as indicated at 1641. The even part of the base-domain image is Jev(n₁,n₂)=ifft₂(MXC). An operation Jev(1,1)=Jev(1,1)−C removes the base-domain form of the real-valued marker, indicated as block 1141. The real part of the transform-domain image is RX=real (fft₂(Jev)).

[0101] Block 1650 calculates the imaginary part of the restored transform-domain image—that is, restores the phase of the original image. Consider the following derivation of the imaginary part IX from MXC, the modified magnitude: $\quad {{{MXC}\hat{}2} = {{{{\left( {\underset{\_}{RX} + \underset{\_}{C}} \right)\hat{}2} + {\underset{\_}{IX}\hat{}2}}\quad\therefore\quad {\underset{\_}{IX}\hat{}2}} = {{{\underset{\_}{MXC}\hat{}2} - {\left( {\underset{\_}{C} + \underset{\_}{RX}} \right)\hat{}2}}\quad = {{\left( {\underset{\_}{MXC} - \left( {\underset{\_}{C} + \underset{\_}{RX}} \right)} \right)*\left( {\underset{\_}{MXC} + \left( {\underset{\_}{C} + \underset{\_}{RX}} \right)} \right)}\quad = {{\left( {\underset{\_}{MXC} - \underset{\_}{C} - \underset{\_}{RX}} \right)*\left( {\underset{\_}{MXC} + \underset{\_}{C} + \underset{\_}{RX}} \right)}\quad = {{{\underset{\_}{C}\hat{}2}*\left( {{\underset{\_}{MXC}/\underset{\_}{C}} - 1 + {\underset{\_}{R\quad X}/\underset{\_}{C}}} \right)*\quad {\left( {{\underset{\_}{MXC}/\underset{\_}{C}} + 1 + {\underset{\_}{RX}/\underset{\_}{C}}} \right).\quad {Defining}}\quad \underset{\_}{AN}} \equiv {{\underset{\_}{MXC}/\underset{\_}{C}} - 1 + {{\underset{\_}{RX}/\underset{\_}{C}}\quad {and}}}}}}}}}$ $\quad {{{AP} \equiv {{\underset{\_}{MXC}/\underset{\_}{C}} + 1 + {\underset{\_}{RX}/\underset{\_}{C}}}},\quad {\underset{\_}{IX} = {\underset{\_}{C}*{sqrt}\quad \left( \underset{\_}{AN} \right)*{sqrt}\quad \left( \underset{\_}{AP} \right)}}}$

[0102] This computation differs somewhat from earlier phase restoration in order to retain accuracy of the element values in view of precision limitations in the data processor, because MXC and C are very large numbers. AN should have all positive values. Block 1650 sets any slightly negative values due to precision shortcomings are to zero by AN=(AN>0).*AN. The correct (bipolar) version of IX is not needed in this example. If desired, the algebraic signs of the IX elements may be corrected by reintroducing the sign map, noted at block 1661. The correction is IX=IX.* RMAP.* MAP₁, where RMAP=1−2* (RX<0). The purpose of RMAP is to correct the signs that might have been changed due to precision errors.

[0103] Block 1660 combines the real and imaginary parts to find the complete transform-domain restored image. In polar coordinates, the magnitude of the final transform-domain image then becomes MX=sqrt(RX{circumflex over ( )}2+IX{circumflex over ( )}2), and the phase is PX=MAP₁.* arccos (RX/MX). Again, MAP₁ reintroduces the sign-map correction, indicated at block 1661. If Cartesian coordinates Y are desired, the transform-domain image expression is Y(k₁,k₂)=RX(k₁,k₂)+j IX(k₁,k₂).

[0104] Block 1680 takes an inverse discrete Fourier transform to produce the base-domain restored image J(k₁,k₂)=ifft₂(Y). If only the transform-domain restored image is required, block 1680 may be omitted.

[0105] In methods 1500 and 1600, the marker value C had a strictly real value. Modifications to those methods, however, would allow imaginary marker values, generated in the same way as the real values. In block 1550, however, magnitude MXC(k₁,k₂) would become MXC(k₁,k₂)=|X(k₁,k₂)+j C|.

[0106] In method 1600, for an imaginary marker, the inverse transform of the magnitude matrix MXC(k₁,k₂) produces an approximation to the odd portion Jod(n₁,n₂) of the complex image I(n₁,n₂), and the imaginary marker value C is subtracted from the appropriate element of the complex matrix. Where the marker resides in the first pixel, Jod(1,1)=Jod(1,1)−j C. The imaginary part IX(k₁,k₂) of the complex transform becomes IX=−imag (fft₂(Jod)). The matrix j IX(k₁,k₂) is imaginary and bipolar, so a sign map may be calculated as IMAP(k₁,k₂)=1−2 * (IX<0). The real part RX(k₁,k₂) of the of the complex transform matrix X(k₁,k₂) remains the same. Retaining element precision here favors a slightly different computation: BN=MXC/C−1+IX/C. Again, any slightly negative values are set to zero by BN=(BN>0).* BN. BN corresponds to AN, above. BP=MXC/C+1+IX/C. corresponds to AP. The magnitude of the RX(k₁,k₂) may then be computed as RX=C*sqrt(BN).*sqrt(BP). The bipolar matrix version of the RX(k₁,k₂), if needed, is RX=IMAP.*MAP₁ .*RX. Block 1660 would calculate the phase component as PX(k₁,k₂)=MAP₁.*arcsin (IX/MX). Otherwise, the operations may be the same as described in connection with FIGS. 15 and 16.

[0107] Complex image elements, whether transform or base domain, are frequently expressed in single- or even double-precision floating-point notation from the beginning. Therefore in many cases an exact restoration down to the bit level is questionable. However, if the original image elements X(k₁,k₂) are—or can be considered to be—integer values, then the rounding techniques described in connection with methods 400 and 800 may be employed in method 1600 as well. Although the storage-reduction blocks in methods 1500 and 1600 are especially useful in high-precision image processing, they may be incorporated in nay of the other methods as well.

[0108]FIG. 17 shows an application of methods capable of modifying and restoring complex-valued images. Method 1700 modifies, communicates, and restores a pair of independent real-valued images as a single component of a single image. Original images 1711 and 1712 may comprises images of any type, as long as their pixels or other elements do not have complex values. These will be termed “real-valued” pixels, although their elements could be strictly imaginary or represent any other single value. They are symbolized herein as I1(n₁,n₂) and I2(n₁,n₂).

[0109] Block 1720 converts one of the images to have imaginary pixel values, j*I2(n₁,n₂). Block 1730 then combines the images into a single image having complex pixel values, I(n₁,n₂)=I1(n₁,n₂)+j*I2(n₁,n₂). That is, each pixel of image 1711 become the real part of a pixel of the combined image, and each pixel of image 1712 becomes the complex part of a corresponding pixel in the combined image. These terms may include Cartesian angles other than 0° and 90°, or magnitude and phase, or any other pair of mutually orthogonal complex representations.

[0110] Block 1740 modifies the combined image I(n₁,n₂) to produce a single component (magnitude or phase) 1750, which has the storage and bandwidth requirements of other modified images described above. This component may be communicated by transmission or storage on a medium such as 130, FIG. 1.

[0111] Block 1760 retrieves the single component and restores it according to methods such as 800 described above to produce the original combined image I(n₁,n₂). Block 1770 separates the restored image into its real and imaginary parts, ReI(n₁,n₂) and ImI(n₁,n₂). Block 1780 reconverts the imaginary part to real values, if necessary, say as I2(n₁,n₂)=ImI(n₁,n₂)/j. Blocks 1791 and 1792 represent the restored images I1(n₁,n₂)=ReI(n₁,n₂), and I2(n₁,n₂)=ImI(n₁,n₂)/j.

Conclusion

[0112] Embodiments of the invention offer methods and systems for communicating images efficiently by reducing storage space and transmission time/bandwidth. Images may have any number of dimensions or formats, the term “image” herein is not limited to spatial images, and may include any sequence of data that is representable in an appropriate form. Complex or other multi-part element values may be accommodated. Any of the embodiments may process an original image that is presented either in a base domain or in a transform domain. Transforms other than DFT may be employed. The invention may be used along with other compression methods, and is transparent to them.

[0113] The foregoing description and drawing illustrate specific embodiments of the invention sufficiently to enable those skilled in the art to practice it. Other embodiments may incorporate structural, logical, electrical, process, and other changes. Examples merely typify possible variations. Individual components and functions are optional unless explicitly required, and the sequence of operations may vary, or may be performed in parallel. Portions and features of some embodiments may be included in or substituted for those of others. Items in lists are not exclusive, and may also be combined in any way. The Abstract is provided solely as a search aid, and is not to be used for claim interpretation. The scope of the invention encompasses the full ambit of the following claims and all available equivalents. 

1. A method for processing an original image, comprising: selecting a known marker value related to predetermined data in the original image; placing the marker value at a known location with respect to the original image so as to modify certain of the image data; if the original image is in a base domain, transforming it to a multiple complementary components in a transform domain; communicating only one of the components as a modified image; receiving the one component for restoration; generating in a closed form, without iteration a plurality of sequences from the one component; removing the predetermined marker data from the marker location(s); combining the sequences to produce an entire restored image.
 2. The method of claim 1 where the marker value is sufficient to ensure that the component is minimum-phase.
 3. The method of claim 2 where the marker value is complex.
 4. The method of claim 1 where elements of the original image are complex in the base domain.
 5. The method of claim 4 where the original image is a composite of two independent images.
 6. A machine-readable medium for storing program instructions for carrying out the method of claim
 1. 7. A system for processing an original image, comprising: a first digital processor for selecting a known marker value related to predetermined data in the original image, placing the marker value at a known location with respect to the original image so as to modify certain of the image data, if the original image is in a base domain, transforming it to multiple complementary components in a transform domain; a first storage associated with the first processor for storing at least one of the components; a channel for communicating only one of the components; a second digital processor for receiving the one component for restoration, generating in a closed form, without iteration a plurality of sequences from the one component, removing the predetermined marker data from the marker location(s); combining the sequences to produce an entire restored image. a second storage associated with the second processor for storing the received one component
 8. The system of claim 7 where the channel transmits the one component from the first processor to the second processor.
 9. The system of claim 7 where the channel comprises a storage medium for storing the one component.
 10. The system of claim 7 further comprising an input device for producing the original image.
 11. The system of claim 7 further comprising an output device coupled to the second storage for receiving the restored image.
 12. A method for processing image data representing an original image in a base domain for subsequent restoration, comprising: selecting a known marker value related to predetermined data in the original image, where the marker value is not strictly real; placing the marker value at a known location with respect to the original image so as to modify certain of the image data in the base domain; transforming the original image including the modified image data to a transform domain having multiple complementary components; communicating less than all of the components as a modified image.
 13. The method of claim 12 where the marker is placed in the image directly in the base domain.
 14. The method of claim 12 where the marker is placed directly in at least one of the complementary components.
 15. The method of claim 12 where the phase of the marker is at least 1° away from 0°, 90°, 180°, and 270°.
 16. The method of claim 15 where the marker phase is about 45°.
 17. The method of claim 15 where the marker phase is strictly imaginary.
 18. A machine-readable medium for storing program instructions for carrying out the method of claim 12
 19. A system for processing image data representing an original image in a base domain for subsequent restoration, comprising: a processor for selecting a known marker value related to predetermined data in the original image, where the marker value is not strictly real, placing the marker value at a known location with respect to the original image so as to modify certain of the image data in the base domain, transforming the original image including the modified image data to a transform domain having multiple complementary components; a channel for communicating less than all of the components as a modified image.
 20. The system of claim 19 further including a source device for producing the original image.
 21. The system of claim 20 where the source device comprises at least one of a scanner and a storage device. 22 The system of claim 19 where the channel transmits the one component from the processor to a network.
 23. The system of claim 19 where the channel comprises a storage medium for storing the one component.
 24. A method for processing an original image in a transform domain having multiple components for subsequent restoration, comprising: selecting a known marker value in a base domain related to predetermined data for the original image in the base domain, where the marker value is not strictly real; manipulating the original image entirely within the transform domain so as to place data therein representing a transform of the known base-domain marker value at a known location in the base domain; communicating less than all of the components of the manipulated image as a modified image.
 25. The method of claim 24 where the phase of the marker is at least 1° away from 0°, 90°, 180°, and 270°.
 26. The method of claim 25 where the marker phase is about 45°.
 27. The method of claim 25 where the marker phase is strictly imaginary.
 28. A machine-readable medium for storing program instructions for carrying out the method of claim
 24. 29. A system for processing an original image in a transform domain having multiple components for subsequent restoration, comprising: a processor for selecting a known marker value in a base domain related to predetermined data for the original image in the base domain, where the marker value is not strictly real, manipulating the original image entirely within the transform domain so as to place data therein representing a transform of the known base-domain marker value at a known location in the base domain; a channel for communicating less than all of the components of the manipulated image as a modified image.
 30. The system of claim 29 further including a source device for producing the original image.
 31. The system of claim 30 where the source device comprises at least one of a scanner and a storage device. 32 The system of claim 29 where the channel transmits the one component from the processor to a network.
 33. The system of claim 29 where the channel comprises a storage medium for storing the one component.
 34. A method for processing image data representing an original image for subsequent restoration, comprising: selecting a known marker value related to predetermined data of the original image; placing the marker value at a known location with respect to the original image so as to modify certain of the image data; if the original image is not already represented in a transform domain having multiple complementary components, transforming it to produce such components; deriving a map representing a certain characteristic of one of the complementary components; embedding the map within another of the complementary components; communicating only the other of the components, including the embedded map, as a modified image.
 35. The method of claim 34 where the certain characteristic comprises signs of the elements of one component.
 36. The method of claim 35 where embedding the map comprises multiplying elements of the other component by corresponding elements of the map.
 37. A machine-readable medium for storing program instructions for carrying out the method of claim 34
 38. A system for processing image data representing an original image for subsequent restoration, comprising: a processor for selecting a known marker value related to predetermined data of the original image, placing the marker value at a known location with respect to the original image so as to modify certain of the image data, if the original image is not already represented in a transform domain having multiple complementary components, transforming it to produce such components, deriving a map representing a certain characteristic of one of the complementary components, embedding the map within another of the complementary components; a channel for communicating only the other of the components, including the embedded map, as a modified image.
 39. The system of claim 38 further including a source device for producing the original image.
 40. The system of claim 39 where the source device comprises at least one of a scanner and a storage device. 41 The system of claim 38 where the channel transmits the one component from the processor to a network.
 42. The system of claim 38 where the channel comprises a storage medium for storing the one component.
 43. A method for processing image data representing an original image for subsequent restoration, comprising: selecting a known marker value related to predetermined data of the original image; placing the marker value at a known location with respect to the original image so as to modify certain of the image data; if the original image is not already represented in a transform domain having multiple complementary components, transforming it to produce such components; deriving a map representing a certain characteristic of one of the complementary components; communicating the map and the one component as a modified image.
 44. The method of claim 43 further comprising modifying the magnitudes of elements of the one component before deriving the map therefrom.
 45. The method of claim 43 where the characteristic concerns the relationship of the element magnitudes of the one component to a predetermined magnitude.
 46. The method of claim 43 where the characteristic concerns storage amounts required for the magnitudes of individual elements of the one component.
 47. The method of claim 46 further comprising storing the individual elements of the one component in different amounts of storage, depending upon their magnitudes.
 48. A machine-readable medium for storing program instructions for carrying out the method of claim 41
 49. A system for processing image data representing an original image for subsequent restoration, comprising: a digital processor for selecting a known marker value related to predetermined data of the original image, placing the marker value at a known location with respect to the original image so as to modify certain of the image data, if the original image is not already represented in a transform domain having multiple complementary components, transforming it to produce such components; deriving a map representing a certain characteristic of one of the complementary components; a channel for communicating the map and the one component as a modified image.
 50. The system of claim 49 further including a source device for producing the original image.
 51. The system of claim 50 where the source device comprises at least one of a scanner and a storage device. 52 The system of claim 49 where the channel transmits the one component from the processor to a network.
 53. The system of claim 49 where the channel comprises a storage medium for storing the one component.
 54. A method for restoring an image to an original image, comprising: receiving one component of multiple complementary components of the original image, the one component containing predetermined known marker data corresponding to one or more marker locations; generating in a closed form, without iteration, a plurality of sequences from the one component; removing the predetermined marker data from the marker location(s); combining the sequences to produce an entire restored image.
 55. The method of claim 54 where the value of the marker data is sufficient to ensure that the original data is minimum-phase in a transform domain.
 56. The method of claim 54 where the sequences represent even and odd parts of the original image.
 57. The method of claim 54 where the sequences represent real and imaginary parts of the original image.
 58. The method of claim 54 where the sequences represent real and imaginary parts of the original image.
 59. The method of claim 54 where the marker is restricted to a complex value.
 60. The method of claim 54 where the marker has a strictly real or strictly imaginary value.
 61. The method of claim 54 where the sequences are approximate parts of the original image.
 62. The method of claim 61 further comprising converting the sequences from approximate forms to exact forms.
 63. The method of claim 51 where the received one component includes a map having an element for every element of the one component, and further comprising modifying the elements of the one component in accordance with the elements of the map.
 64. The method of claim 63 where the map elements represent signs of elements in a component complementary to the one component.
 65. The method of claim 63 where the map elements represent signs of elements in the one component.
 66. The method of claim 63 where the map elements represent different lengths of individual elements in the one component.
 67. The method of claim 66 further comprising expanding the different element lengths to the same length.
 68. A machine-readable medium for storing program instructions for carrying out the method of claim
 54. 69. A system for restoring an image to an original image, comprising: a channel for receiving one component of multiple complementary components of the original image, the one component containing predetermined known marker data corresponding to one or more marker locations; a digital processor for generating in a closed form, without iteration, a plurality of sequences from the one component, removing the predetermined marker data from the marker location(s), combining the sequences to produce an entire restored image.
 70. The system of claim 69 where the channel comprises an external storage device holding the one component.
 71. The system of claim 69 where the channel is coupled to a network for receiving the one component.
 72. The system of claim 69 further comprising at least one output device coupled to the processor for receiving the restored image.
 73. The system of claim 72 where the output device comprises at least one of a display, a printer, and a data analyzer.
 74. A computer-implemented method comprising: receiving only one component of a transform-domain original image; converting the one component to a base-domain approximate restored image; separating the approximate restored image into even and odd parts; removing the known marker from one of the parts to form a corrected part; producing exact parts from the corrected part and at least one of the other parts; combining the exact parts to produce a final restored image; outputting the final restored image.
 75. The method of claim 74 where the marker value is sufficiently large to guarantee that the original image is minimum-phase in the transform domain.
 76. The method of claim 74 where the one component is a magnitude component of a Fourier transform.
 77. The method of claim 76 where the marker has a complex value.
 78. The method of claim 74 where the transform-domain original image represents a transform of a base-domain original image having rational elements of a predetermined precision.
 79. The method of claim 78 where the final restored image has rational elements equal to the elements of the base-domain original image, within the predetermined precision.
 80. The method of claim 74 further comprising transforming the exact parts to produce a transform-domain final restored image.
 81. A machine-readable medium for storing program instructions for carrying out the method of claim 74
 82. A system comprising: a channel for receiving only one component of a transform-domain original image; a digital processor for converting the one component to a base-domain approximate restored image, separating the approximate restored image into even and odd parts, removing the known marker from one of the parts to form a corrected part, producing exact parts from the corrected part and at least one of the other parts, combining the exact parts to produce a final restored image, outputting the final restored image.
 83. The system of claim 82 where the channel comprises an external storage device for holding the one component.
 84. The system of claim 82 where the channel is coupled to a network for receiving the one component.
 85. The system of claim 82 further comprising at least one output device coupled to the processor for receiving the restored image.
 86. The system of claim 85 where the output device comprises at least one of a display, a printer, and a data analyzer.
 87. A computer-implemented method comprising: receiving only one component of a transform-domain original image; forming a complex transform from the one component; removing a predetermined marker value to produce a corrected restored image; calculating a real part of the corrected restored image.
 88. The method of claim 87 where the one component is a phase component of a Fourier transform
 89. The method of claim 87 where the marker value is sufficiently large to guarantee that the original image is minimum-phase in the transform domain.
 90. The method of claim 89 where the marker has a complex value.
 91. The method of claim 87 where removing the marker includes inverse-transforming the complex transform to a base-domain restored image; removing a marker value from the base-domain restored image so as to change at least one element of the base-domain restored image.
 92. The method of claim 91 where the marker is removed by subtracting it from a predetermined element of the base-domain restored image.
 93. The method of claim 87 further comprising producing an exact restored image from the corrected restored image.
 94. The method of claim 93 where the exact restored image is produced by rounding a function of the corrected restored image.
 95. A machine-readable medium for storing program instructions for carrying out the method of claim
 87. 96. A system comprising: a channel for receiving only one component of a transform-domain original image; a digital processor for forming a complex transform from the one component, removing a predetermined marker value to produce a corrected restored image, calculating a real part of the corrected restored image.
 97. The system of claim 93 where the channel comprises an external storage device for holding the one component.
 98. The system of claim 93 where the channel is coupled to a network for receiving the one component.
 99. The system of claim 93 further comprising at least one output device coupled to the processor for receiving the restored image.
 100. The system of claim 93 where the output device comprises at least one of a display, a printer, and a data analyzer.
 101. A computer-implemented method comprising: receiving only one component of a transform-domain original image, the one component including a sign map; extracting the sign map from the one component; calculating a magnitude of the one component; restoring another component of the transform-domain image in response to the sign map; inverse-transforming the components to produce a base-domain restored image; removing a predetermined marker value from the restored image.
 102. The method of claim 101 further comprising: expanding the resolution of the one component; producing a final image from the restored image.
 103. The method of claim 101 where the transform-domain original image represents a transform of a base-domain original image having elements of a predetermined precision.
 104. The method of claim 103 where the final restored image has rational elements equal to the elements of the base-domain original image, within the predetermined precision.
 105. The method of claim 101 where the marker value ensures that the transform-domain original image is minimum-phase.
 106. The method of claim 105 where the marker value is real.
 107. The method of claim 101 where the sign map has elements representing the signs of elements of another transform-domain component of the original image.
 108. The method of claim 107 further comprising embedding the sign map within the one component.
 109. The method of claim 108 where the sign map is embedded by multiplying its elements by corresponding elements of the one compionent.
 110. The method of claim 107 where extracting the sign map includes detecting signs of elements of the one component.
 111. A machine-readable medium for storing program instructions for carrying out the method of claim
 101. 112. A system comprising: a channel for receiving only one component of a transform-domain original image, the one component including a sign map; a digital processor for extracting the sign map from the one component; calculating a magnitude of the one component; restoring another component of the transform-domain image in response to the sign map; inverse-transforming the components to produce a base-domain restored image; removing a predetermined marker value from the restored image.
 113. The system of claim 112 where the channel comprises an external storage device for holding the one component.
 114. The system of claim 112 where the channel is coupled to a network for receiving the one component.
 115. The system of claim 112 further comprising at least one output device coupled to the processor for receiving the restored image.
 116. The system of claim 115 where the output device comprises at least one of a display, a printer, and a data analyzer.
 117. A computer-implemented method comprising: receiving only one component of a transform-domain original image, the one component comprising multiple elements; receiving a location map having elements designating storage lengths for the elements of the one component; reconstituting the one component in response to the location map; calculating real and imaginary parts of a restored image from the reconstituted component; removing a predetermined marker value from at least one of the parts; combining the parts to produce the restored image.
 118. The method of claim 117 further comprising receiving a magnitude-sign map having elements indicating the relationships of respective elements of the one component to a threshold magnitude.
 119. The method of claim 118 where the threshold magnitude is zero.
 120. The method of claim 118 where the one component is further reconstituted in response to the magnitude-sign map.
 121. The method of claim 117 further comprising inverse-transforming the restored image to a base-domain final image.
 122. The method of claim 121 where the final image has multiple elements each having first and second numbers.
 123. The method of claim 122 where the two numbers of each final-image element comprise a real number and an imaginary number.
 124. The method of claim 122 further comprising separating the final image into two separated final images.
 125. The method of claim 124 where each of the separated images compresses a respective one of the numbers of each element of the final image.
 126. The method of claim 117 where the parts are real and imaginary parts of the restored image in the transform domain.
 127. The method of claim 126 further comprising inverse transforming the restored image to a base domain.
 128. The method of claim 117 where the marker value is sufficient to ensure minimum phase of the transform-domain original image.
 129. The method of claim 128 where the marker value is real.
 130. The method of claim 128 where the marker value is imaginary.
 131. A machine-readable medium for storing program instructions for carrying out the method of claim 117
 132. A system comprising: a channel for receiving only one component of a transform-domain original image, the one component comprising multiple elements; a digital processor for receiving a location map having elements designating storage lengths for the elements of the one component, reconstituting the one component in response to the location map, calculating real and imaginary parts of a restored image from the reconstituted component, removing a predetermined marker value from at least one of the parts, combining the parts to produce the restored image.
 133. The system of claim 132 where the channel comprises an external storage device for holding the one component.
 134. The system of claim 132 where the channel is coupled to a network for receiving the one component.
 135. The system of claim 132 further comprising at least one output device coupled to the processor for receiving the restored image.
 136. The system of claim 135 where the output device comprises at least one of a display, a printer, and a data analyzer.
 137. A method of processing image data. comprising: receiving at least first and second original images each comprising multiple elements each having a single value; converting the elements of at least the first original image to different values; combining the first and second images to a composite original image having multiple elements each having a pair of orthogonal values; placing a known marker value related to predetermined data of the composite original image in the composite image so as to modify certain of the composite-image data; transforming the composite image including the modified data to a transform-domain composite original image having multiple complementary components; communicating only one of the components as a modified composite image.
 138. The method of claim 137 where the marker value is sufficient to ensure that the composite image is minimum-phase in the transform domain.
 139. The method of claim 137 where the original images are sequences of values in a base domain.
 140. The method of claim 137 where the orthogonal values represent real and imaginary parts of complex numbers.
 141. The method of claim 140 where converting the first original image comprises representing its elements as imaginary numbers.
 142. The method of claim 140 where the elements of the second original image are represented as real numbers.
 143. A machine-readable medium for storing program instructions for carrying out the method of claim
 137. 144. A system for processing image data. comprising: a digital processor for receiving at least first and second original images each comprising multiple elements each having a single value, converting the elements of at least the first original image to different values, combining the first and second images to a composite original image having multiple elements each having a pair of orthogonal values, placing a known marker value related to predetermined data of the composite original image in the composite image so as to modify certain of the composite-image data, transforming the composite image including the modified data to a transform-domain composite original image having multiple complementary components; a channel for communicating only one of the components as a modified composite image.
 145. The system of claim 144 further including a source device for producing the original image.
 146. The system of claim 145 where the source device comprises at least one of a scanner and a storage device. 147 The system of claim 144 where the channel transmits the one component from the processor to a network.
 148. The system of claim 144 where the channel comprises a storage medium for storing the one component.
 149. A method of processing image data. comprising: receiving only one component of a transform-domain modified composite image, representing two separate single-valued original images as multiple-valued elements each containing orthogonal values from both of the original images; restoring the composite image from the one component so as to produce a composite restored image having elements each having multiple orthogonal values. removing a predetermined marker value so as to change the data of at least one element of the restored composite image; separating the composite restored image into two individual restored images each having multiple elements having only one value per element, each representing one of the single-valued original images.
 150. The method of claim 149 where the marker value is sufficient to ensure that the composite image is minimum-phase in the transform domain.
 151. The method of claim 149 where the orthogonal values represent real and imaginary parts of complex numbers.
 152. The method of claim 151 further comprising converting one of the individual restored images from imaginary values to real numbers.
 153. The method of claim 151 where the elements of the second restored image are represented as real numbers.
 154. The method of claim 149 where the restored images are sequences of values in a base domain.
 155. A machine-readable medium for storing program instructions for carrying out the method of claim
 149. 156. A system for processing image data. comprising: a channel for receiving only one component of a transform-domain modified composite image, representing two separate single-valued original images as multiple-valued elements each containing orthogonal values from both of the original images, a digital processor for restoring the composite image from the one component so as to produce a composite restored image having elements each having multiple orthogonal values, removing a predetermined marker value so as to change the data of at least one element of the restored composite image, separating the composite restored image into two individual restored images each having multiple elements having only one value per element, each representing one of the single-valued original images.
 157. The system of claim 156 where the channel comprises an external storage device for holding the one component.
 158. The system of claim 156 where the channel is coupled to a network for receiving the one component.
 159. The system of claim 156 further comprising at least one output device coupled to the processor for receiving the restored image.
 160. The system of claim 159 where the output device comprises at least one of a display, a printer, and a data analyzer. 